Rangaprakash et al. Brain Inf. (2020) 7:19 https://doi.org/10.1186/s40708-020-00120-2 Brain Informatics

RESEARCH Open Access Density‑based clustering of static and dynamic functional MRI connectivity features obtained from subjects with cognitive impairment D. Rangaprakash1,2,3, Toluwanimi Odemuyiwa4, D. Narayana Dutt5, Gopikrishna Deshpande6,7,8,9,10,11,12,13* and Alzheimer’s Disease Neuroimaging Initiative

Abstract Various machine-learning classifcation techniques have been employed previously to classify brain states in healthy and disease populations using functional magnetic resonance imaging (fMRI). These methods generally use super- vised classifers that are sensitive to outliers and require labeling of training data to generate a predictive model. Density-based clustering, which overcomes these issues, is a popular approach whose util- ity for high-dimensional neuroimaging data has not been previously evaluated. Its advantages include insensitivity to outliers and ability to work with unlabeled data. Unlike the popular k-means clustering, the number of clusters need not be specifed. In this study, we compare the performance of two popular density-based clustering methods, DBSCAN and OPTICS, in accurately identifying individuals with three stages of cognitive impairment, including Alzhei- mer’s disease. We used static and dynamic functional connectivity features for clustering, which captures the strength and temporal variation of brain connectivity respectively. To assess the robustness of clustering to noise/outliers, we propose a novel method called recursive-clustering using additive-noise (R-CLAN). Results demonstrated that both clustering algorithms were efective, although OPTICS with dynamic connectivity features outperformed in terms of cluster purity (95.46%) and robustness to noise/outliers. This study demonstrates that density-based clustering can accurately and robustly identify diagnostic classes in an unsupervised way using brain connectivity. Keywords: Functional MRI, Brain networks and dynamic connectivity, Cognitive impairment and alzheimer’s disease, Unsupervised learning and clustering, DBSCAN, OPTICS

1 Introduction data? In response to this, classifers Since the successful emergence of functional neuroimag- have been employed on neuroimaging features to gener- ing, a new barrier has surfaced: can a strong correlation ate models that, within some accuracy, predict the cogni- be established between brain activity (as measured by tive and disease states to which new data belong [1–11]. functional magnetic resonance imaging [fMRI]) and the Tere are two major categories of machine learning cognitive state of an individual? More specifcally, can we classifcation techniques: supervised and unsupervised. accurately classify neurological diseases based on fMRI , commonly used in fMRI studies, involves splitting the dataset into training and test data. *Correspondence: [email protected] Each member of the training data is given a ‘label’ as to 6 AU MRI Research Center, Department of Electrical and Computer which class (or group) it belongs to. Te classifer is then Engineering, Auburn University, 560 Devall Dr, Suite 266D, Auburn, AL ‘trained’ on this data to determine a generalized model 36849, USA Full list of author information is available at the end of the article (thus ‘supervised’ learning). Ten, the classifcation

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accuracy is measured by testing the model on the test demonstrate the utility and feasibility of the unsupervised data with known labels. Conversely, in unsupervised learning approach of density-based clustering, applied on learning, patterns within the entire dataset are used to resting-state fMRI (rs-fMRI) data to cluster disease states ‘cluster’ the data without any pre-assigned labels, and in a noise-robust manner. cluster purity is measured against the known ground- Clustering is an important technique of grouping truth, post hoc, instead of an accuracy. As such, unsuper- objects that are similar to each other and isolating them vised learning is agnostic to pre-assigned labels, and thus from those that are dissimilar. Clusters are formed with determines inherent classes instead of ftting a model features having minimum intra-cluster distance and based on classes provided by us. Although the ‘cluster maximum inter-cluster distance [28]. Clustering has purity’ given by clustering is, in principle, the same as found wide application in many felds including, but not the ‘classifcation accuracy’ given by supervised classif- limited to, the recovery of information, recognition of ers (both give the percentage of correct classifcations as natural patterns, the analysis of digital images, bioinfor- against the known ground truth), we would use the term matics, , taxonomy, DNA microarray analy- ‘cluster purity’ in this work, so as to highlight that this sis, and many others [29–34]. Te performance of most metric was obtained through unsupervised clustering, clustering algorithms, however, is highly sensitive to and not through conventional supervised classifcation. their input parameters, such as the number of clusters in FMRI studies generally use supervised learning meth- k-means clustering [35]. For efective performance, one ods to classify disease or cognitive states [1–3, 12, 13]. almost needs a priori knowledge of these parameters. Unsupervised learning methods are generally used for Although there are a variety of clustering algorithms, spatially and temporally clustering voxel signals to local- density-based clustering is one of the most computation- ize brain activity [10, 11, 14–20]; for dimensionality ally efective methods in clustering large-scale databases reduction or before applying a super- [36]. Unlike k-means clustering, density-based clustering vised learning algorithm to the data [13, 21–24]; or for techniques do not need the user to specify the number mapping specifc brain patterns to a cognitive state [6, of clusters since the algorithm determines that by itself. 25]. A few reports have adopted unsupervised learning Using a local proximity measure, clusters are automati- methods as state classifers across individuals. For exam- cally formed by points in higher density regions, well sep- ple, one study used the one-class support vector machine arated by lower density regions. In this work, we focus on (SVM) to determine the boundary around healthy con- two of the most popular density-based clustering meth- trol subjects and to classify outliers based on their dis- ods: Density-Based Spatial Clustering of Applications tance from this boundary. However, such methods are with Noise (DBSCAN) and Ordering Points to Identify highly susceptible to noise and do not work well with the the Clustering Structure (OPTICS). Tey require few high-dimensional data common in fMRI [26]. input parameters, need no a priori user choices, detect Research on clustering of subjects using unsupervised outliers, and efectively detect clusters of arbitrary shapes learning methods is still limited in fMRI literature, to [37, 38]. the best of our knowledge. Diagnostic labels are often DBSCAN is a single-scan technique that makes no pre- predetermined in neuroimaging studies, and most stud- sumptions regarding the distribution of data. In addition ies often stick to this labeling, because of which super- to clustering the data, the algorithm also detects outliers. vised classifers are more popular. However, the main Te OPTICS algorithm was developed by Ankerst et al. advantage of unsupervised learning is that it requires [38] to address DBSCAN’s major shortcoming that it can- no a priori knowledge of categorical labels, thus reduc- not detect clusters with diferent local densities. OPTICS ing the problem of selection bias described by Demerci is a non-rational, data-independent representation of et al. [5]. Tis allows for the learning of more complex the cluster structure that displays important informa- models, pattern discrimination and the identifcation of tion regarding the distribution of the data. It is capable of hidden states [27] among others. Unsupervised learn- identifying all possible clusters of varying shapes. Advan- ing also does not have the issue of overftting the data, tages of OPTICS over DBSCAN are that it eliminates which largely plagues supervised learning models. Te free parameter choices needed in DBSCAN, is relatively overftting nature of supervised learning models results insensitive to noise and can detect clusters of diferent in high performances on the data being used, but much densities [38]. lower performance when tested on an independent data- In fMRI data, noise is any signal variation that is not set. Unsupervised learning does not encounter this issue contributed by neuronal activity. It may arise from head from the very nature of its formulation since learning movements during scanning, scanner limitations, ther- is done in an entirely blind manner. With unsupervised mal noise, or other sources [39, 40]. Te signal-to-noise learning, what you see is what you get. In this work, we ratio (SNR) is a measure of how much signal is present Rangaprakash et al. Brain Inf. (2020) 7:19 Page 3 of 13

in the data relative to noise. Higher SNR values indicate approach yields impressive and reliable results, it would higher quality data with low amounts of noise. However, advance clinical diagnosis and classifcation of cognitive much progress has been made in detecting and minimiz- impairment. ing noise artifacts from fMRI data, like high-pass flter- In this work, we assessed the performance of DBSCAN ing, motion correction and other pre-processing steps and OPTICS, in combination with SFC and DFC fea- [39, 41]. Some noise is always prevalent in the data. It tures, in accurately clustering three groups of cogni- is important that learning methods perform well even tive impairment (including Alzheimer’s disease) and a when given lower quality data. In this work, we assess the matched healthy control group. Higher clustering per- robustness of clustering to such noisy variations. formance need not necessarily imply higher robustness Resting-state fMRI (rs-fMRI) involves the collection of to noisy variations in the data. It is possible that a high fMRI data from subjects while at rest in the scanner, with cluster purity is obtained, but that high performance is eyes open and mind left to wander. It is characterized not sustained if noise in the data increases. Noise robust- by correlations across various regions of the brain, also ness is very important for two reasons: (i) biomedical called functional connectivity (FC). FC estimated from data, especially brain imaging data, are vulnerable to rs-fMRI is extensively used to study brain networks, and a large number of unknown sources of variability and prior works have identifed alterations in FC in subjects known sources of noise, and (ii) the data used in any with various psychiatric and neurological conditions study is only a representative sample of the general popu- (please see [42–45] for a review). We have used func- lation, and hence generalizability and inter-subject vari- tional connectivity features in this work. ability are imminent issues; therefore, if the results have Static Functional Connectivity (SFC) refers to the to be applicable in a real-world setting, then it has to be strength of connectivity between two brain regions and is robust to such variability. Tus, to assess robustness, we quantifed using Pearson’s correlation coefcient between developed a novel technique called ‘recursive clustering pairs of fMRI time series. It evaluates the temporal corre- using additive noise’ (R-CLAN). Additionally, since well- lation between fMRI time series of two brain regions, giv- formed clusters are partly defned by how well the clus- ing one correlation value over the entire duration of the ters are separated in the feature space, we propose the scan. On the other hand, variance of Dynamic Functional ‘separation index’ to supplement the R-CLAN outcome, Connectivity (DFC) [46–48] captures time-varying con- as a measure of OPTICS’ performance. nectivity and is obtained using sliding-window Pearson’s Te organization of the paper is as follows: section-2 correlation between pairs of brain regions. Although SFC describes the methods used for grouping the dataset is the most widely used measure of fMRI connectivity, into clusters and our methods to assess the algorithms’ DFC is emerging as an important measure having cer- robustness; section-3 presents the results with clustering tain unique special properties [42]. SFC and DFC pro- performances and their robustness; section-4 evaluates vide characteristically diferent information regarding and discusses the fndings, and highlights possible future the relationship between two brain regions. While SFC work; and section-5 provides concluding remarks. gives the strength of connectivity or co-activation, DFC gives the variation of connectivity over time (please refer 2 Methods to Hutchison et al. [42] for a review of dynamic connec- 2.1 Data acquisition and pre‑processing tivity). In this work, in addition to evaluating the perfor- Te fMRI data used in this work were obtained from the mance of DBSCAN and OPTICS clustering approaches, Alzheimer’s Disease Neuroimaging Initiative (ADNI) we compared the performance of SFC and DFC features. database (https​://www.adni-info.org/). ADNI is a mul- We explore which of these two popular measures is more tisite, longitudinal study employing imaging, clinical, suitable for clustering in cognitive impairment. bio-specimen and genetic biomarkers in healthy elders In this work, SFC and DFC features were obtained from as well as in individuals with early mild cognitive impair- 132 subjects with progressive stages of cognitive impair- ment (EMCI), late MCI (LMCI) and Alzheimer’s disease ment (early mild cognitive impairment [MCI], late MCI (AD). Te ADNI was launched as a public–private part- and Alzheimer’s disease), along with matched healthy nership, led by Principal Investigator Michael W. Weiner, controls. Cognitive impairment [49] is a spectrum dis- MD. Te primary goal of ADNI has been to test whether order ranging from mild to severe symptoms. Tere serial neuroimaging and other biological markers can be has been enormous interest in identifying subgroups combined with clinical and neuropsychological assess- of cognitive impairment representing relatively homo- ment to measure the progression of MCI and AD. For geneous symptoms [49]. In this work, we performed up-to-date detailed information, please see www.adni- clustering and assessed how well the clustered subject info.org. Table 1 provides the demographics of the sub- groups matched with clinically diagnosed groups. If this jects used by us. Rangaprakash et al. Brain Inf. (2020) 7:19 Page 4 of 13

Table 1 Basic demographics because we did not want the diference in number of fea- Variable Control Early MCI Late MCI AD tures (between SFC and DFC) to impact clustering per- formance. DFC was obtained using sliding-windowed Age, years Pearson’s correlation [46–48], giving a time series of cor- Mean 74.5 72.2 71.4 73.1 relation values. Te variance of these values over time is Median 73.8 72.9 72.3 74.5 a measure of how much the connectivity varies over the SD 5.9 5.9 8.6 7.4 duration of the scan, which was the measure we used fur- Range 20.5 26.8 30.9 30.6 ther. Te width of the sliding windows was not chosen Gender, no. of subjects arbitrarily, like in most studies, but was instead evaluated Male 15 18 21 13 objectively wherein the window lengths were determined Female 21 16 13 16 adaptively using the augmented Dickey-Fuller (ADF) test [46–48], an analytical test based on timeseries station- arity. Overlapping windows were used, with successive windows difering by one time point. Similar to SFC, a A total of 132 subjects were considered from phase-1 132 100 variance of DFC matrix was obtained, which of the database: 35 control subjects, 34 EMCI, 34 LMCI × was then used in clustering. Please refer to Additional and 29 AD patients. RS-fMRI data were acquired in fle 1: Tables S1 through S3 for the list of all the top-100 3 T Philips MR scanners using a T2* weighted sin- signifcant SFC and DFC connectivity paths from which gle shot echo-planar imaging (EPI) sequence with 48 the features were extracted for further analysis. slices and the following parameters: repetition time (TR) 3000 ms, echo time 30 ms, fip angle 80°, voxel = = 3 = 2.3 The DBSCAN algorithm size = 3.3125 × 3.3125 × 3.3125 ­mm , and 140 temporal volumes in each run. Field of view parameters were the For implementation of DBSCAN and OPTICS, we wrote cus- tom Matlab scripts using Daszykowski’s functions (https​:// following: Right–Left (RL) = 212 mm, Anterior–Poste- www.chemo​metri​a.us.edu.pl/index​.php?goto downl​oads). rior (AP) = 198.75 mm, and Foot-Head (FH) = 159 mm. = Anatomical images were acquired using magnetization- Here, we give a brief overview of the DBSCAN algorithm, as prepared rapid acquisition gradient echo (MPRAGE) described by Ester et al. [37]. Te idea, in its simplest form, is that the density of points within a cluster must exceed a cer- sequence for overlay and localization (TR = 6.8 ms, echo 3 tain global threshold. Regions of density below this threshold time = 3.1 ms, voxel size: 1.11 × 1.11 × 1.2 ­mm , fip are considered as noise or outliers. Two input parameters, ε, angle = 9°, feld of view: RL = 204 mm, AP = 253 mm, the minimum radius of a cluster, and MinPts, the minimum FH = 270 mm). Prior to pre-processing, frst fve volumes of the fMRI number of points required in a cluster, form the global thresh- time series were discarded to allow for MR scanner equi- old measures. In distinguishing regions of data by density, the libration. Standard rs-fMRI data preprocessing steps following terms are used: were performed on the raw data (realignment, normali- zation to MNI space, detrending, regressing out nuisance a. ε-neighborhood: For a point p in the cluster, its covariates [6 head motion parameters, white-matter sig- ε-neighborhood is the set of points contained within nal and cerebrospinal fuid signal] using SPM8 [50] and the radius ε from that point. DPARSF [51] toolboxes in the ­Matlab® environment. b. Core point: A point is a core point if its Mean fMRI time series were then extracted from 200 ε-neighborhood contains at least MinPts points. functionally homogeneous regions-of-interest (ROIs) c. Border point: A point is a border point if there are less obtained through (Craddock-200 than MinPts number of points in its ε-neighborhood. atlas, [52]). d. Density-reachable: A point q is directly density- reachable from point p if p is a core point, and q is 2.2 Obtaining connectivity features within the ε-neighborhood of p. In general, q is SFC was obtained between all pair-wise ROIs using density-reachable from p if there is a chain of points from p to q such that each successive point is directly Pearson’s correlation coefcient, giving a 200 × 200 SFC matrix per subject. Multivariate N-way analysis of covari- density-reachable from the previous point. ance (MANCOVAN) was used to perform statistical tests e. Density-connected: Two points, p and q, are density- between SFC features of all groups, controlling for age, connected if there exists a third point from which gender and head motion. Te top-100 (i.e. lowest p val- both are density-reachable. ues) signifcant features were selected for further analysis, that is, a 132 × 100 (subjects × features) SFC matrix was Starting at a core point p, DBSCAN gathers all the used for clustering. We chose a fxed number of features, density-reachable points of p into a single cluster C. Te Rangaprakash et al. Brain Inf. (2020) 7:19 Page 5 of 13

procedure is then repeated again with all the new core 2.5 The OPTICS algorithm points in C until the cluster is completely surrounded Here, we provide a brief overview of OPTICS as by border points from which no other point is density- described by Ankerst et al. [38]. OPTICS is efectively an reachable. DBSCAN then repeats the process again for extension of the DBSCAN algorithm that uses a range of any unclustered points in the database until all points ε values, rather than a single, global threshold, to iden- have been processed. Finally, the points within a cluster tify clusters of diferent local densities. Akin to DBSCAN, will all be density-connected with each other, and the OPTICS uses input parameters MinPts and ‘generat- points lying completely outside these clusters will be con- ing distance’ ε; however, ε is now a maximum threshold sidered as outliers. value. We use ε′ to denote the range of radius values used by OPTICS, where 0 < ε′ < ε. To understand the algo- 2.4 Determination of radius in DBSCAN algorithm rithm, two terms are defned in addition to those used in As with the other clustering algorithms, DBSCAN is sen- DBSCAN: sitive to its input parameters, particularly to ε. As noted in Tench et al. [53], if ε is too large, the algorithm will a. Core distance: Te core distance of a point p is the not be as discriminative and may include outliers in its distance ε′ between p and its MinPts’ neighbor such clusters, and if too small, clusters may not be detected at that p is a core point with respect to ε′ . If this ε′ value all. Consequently, Ester et al. [37] proposed the method is greater than ε, then the core-distance is undefned. of k-distance graphs to determine ε. In this method, all b. Reachability distance (of p from s): If s is not a core the data points are ordered according to their distance d point with respect to ε, the reachability distance is undefned. Otherwise, the reachability distance of from their kth nearest neighbor (setting k = MinPts), and a k-distance graph is formed by plotting d against the a point p from a point s is the maximum of s’s core ordered points in the feature space. By visual inspection, distance and smallest distance such that p is density- ε is chosen to be the value d at which an ‘elbow’ occurs in reachable from s. the plot. However, this method does not provide depend- able ε values [54], although it efectively identifes the OPTICS works on the principle that to accurately dis- range of values where one can search for a more optimal tinguish between diferent densities of clusters; higher ε value. density clusters must frst be found before lower den- We modifed their approach by frst obtaining an ini- sity clusters are identifed. Hence, OPTICS orders data tial estimate of ε using the k-distance method, and then points based on increasing εˈ value, since smaller ε values performing DBSCAN for all ε values ranging from zero indicate higher density regions. Tis ordered list is part to twice this initial estimate, resulting in diferent num- of the output of OPTICS. As in DBSCAN, points not ber of clusters for diferent ε values. Finally, we searched belonging to any cluster are considered outliers. Unlike for the number of clusters that spanned the largest DBSCAN, however, OPTICS does not explicitly assign range of ε values, since this would identify the number cluster memberships; rather, points are ordered based of clusters that were most stable among all choices of on their reachability and core distance values. In addi- the parameter ε. We assigned the mean ε value within tion, OPTICS is relatively insensitive to ε and MinPts. this range as the fnal ε value for performing DBSCAN As long as a large enough value is chosen for ε, and a clustering. Te rationale behind this approach was that choice of around 10–20 is made for MinPts (according to those number of clusters that spanned the largest ε [38]), consistent cluster structures will be identifed. Tis range would represent the most stable and true separa- makes implementing OPTICS practically non-arbitrary tion between the clusters. and choice-free. Tis process can be visualized through what we term as the epsilon plot (Fig. 3, shown later in the results sec- 2.6 Separation index to evaluate robustness tion), which is a graph of the number of clusters identi- of reachability plot in OPTICS clustering fed by DBSCAN against the range of ε values. Since the To assess the clustering structure in OPTICS, a reachabil- number of clusters is a discrete set of values, the plot ity plot is popularly used [38]. Te reachability distance is characterized by a series of horizontal steps, where is plotted for each data point, ordered according to the a step corresponds to the set of ε values that give rise ordered-list output of OPTICS. Figure 1 gives an example to that number of clusters. As described earlier, the ε illustration of the reachability plot defned by four clus- value corresponding to the midpoint of the widest step ters. A series of peaks and valleys characterize the plot, was chosen as the fnal ε value used in the DBSCAN where a valley indicates a cluster. Tis plot efectively clustering. indicates the cluster membership of each point, although Rangaprakash et al. Brain Inf. (2020) 7:19 Page 6 of 13

2.7 Assessing robustness using additive noise Clustering could be defned as robust if it can maintain its performance in the presence of higher noise levels com- pared to noiseless data. Tus, robustness is a measure of how well the clustering would be performed if lower quality data were given. A clustering method providing higher cluster purity does not necessarily imply that it is more robust to noisy variations in the data. It is possible for a clustering method to provide high performance yet be more sensitive to outliers and noise. To assess the robustness of DBSCAN and OPTICS, each in association with SFC and DFC input features, we devised a novel approach called Recursive CLustering using Additive Noise (R-CLAN). Here, white Gaussian Fig. 1 Example illustration of the reachability plot obtained in noise was successively added to SFC (or DFC) features OPTICS such that the SNR value of the subsequent corrupted features decreased by 1 dB per iteration, with SNR val- ues starting from 100 dB in the frst iteration. In this further steps are performed in OPTICS to formally defne context, the traditional defnition of SNR was used, that each cluster. Further details can be found in [38]. Since is, SNR = 10 × log(S/N), where S was the signal power obtaining the reachability plot is an intermediate step in (mean squared value of all the original SFC [or DFC] OPTICS clustering, the success of the method critically values), N was the noise power (mean squared value of depends on the reachability plot. Consequently, assessing additive Gaussian noise), and the logarithm being taken the robustness of the plot is indirectly an assessment of to the base of 10. An SNR value of 100 dB indicates a high the outcome of OPTICS clustering. amount of signal relative to noise (higher quality data), To assess the performance of OPTICS on both whereas a value of 0 dB indicates an equal amount of SFC and DFC features using the reachability plot, we noise and signal in the data, making it difcult to distin- devised a novel measure called the OPTICS separation guish signal from noise (lower quality data). index. Leaving out the frst and last points of the reach- Starting with data having an SNR of 100 dB, we per- ability plot, we defned the separation index as the ratio formed DBSCAN and OPTICS clustering at each SNR of the mean of peak heights to the mean value of points value, decreasing the SNR by 1 dB in each iteration and between the peaks in the valleys. For example, suppose terminating only when the clustering structure had there exists peaks at points i and j in the reachability changed (i.e. the cluster purity value changed from the plot, with cluster-k defned by the valley bounded by original at 100 dB). Tat means, we stopped at that SNR these peaks, then the separation index for cluster-k is value when at least one subject in one of the groups was defned as: clustered into another group, thus changing the groups’ 1 RP(i) + RP j structure. At this point, the terminating SNR value was = 2  , Sk −1 taken as a measure of robustness. Te lower the SNR 1 j ( ) (1) j−i−1 m=i+1 RP m value, the more noise that was present in the data, thus more robust the algorithm would be in combination with where Sk is the separation index for the kth group, and the input features (SFC or DFC), since the low SNR indi- RP(i), RP(j) are the values of the reachability plot. Te cates a higher tolerance for noise. Figure 2 provides the fnal separation index is an average of the separation fowchart for this procedure. R-CLAN and separation- indices obtained for each cluster. In simple terms, the index procedures complement one another and were separation index evaluates the ratio of the peak heights to used to measure robustness objectively. the average of the baseline heights. A higher separation index value indicates that the peaks are relatively much 3 Results larger than the baseline; hence, obtaining clusters using We frst present the fndings from DBSCAN and OPTICS a threshold on the reachability plot would then be less clustering, before reporting the results from the assess- prone to noise from outliers. Tis is an indirect measure ment of robustness using our R-CLAN and separation of the robustness of OPTICS clustering. index techniques. Figure 3a, b shows the epsilon plots Rangaprakash et al. Brain Inf. (2020) 7:19 Page 7 of 13

generated using DBSCAN with SFC and DFC features, respectively. With both plots, our method of choosing ε resulted in DBSCAN detecting a total of four clusters, which was the expected number, given that we had four diagnostic groups. Clustering performances are presented in Table 2. With DBSCAN clustering, the average group-wise cluster purity with SFC and DFC features were 75% and 87.88%, respectively; while OPTICS clustered with 93.18% cluster purity using SFC and 95.46% using DFC features. From this, it is clear that, (i) OPTICS performed better than DBSCAN, (ii) DFC features resulted in higher perfor- mance than SFC features, (iii) OPTICS clustering using DFC features resulted in the overall best performance, and (iv) the performance with control and AD groups were higher than that with the intermediate EMCI and LMCI groups. Te execution times for a single instance of DBSCAN and OPTICS algorithms on our computer (Intel­ © Xeon­ © quad-core processor, 3.5 GHz) were 0.0206 s and 0.0204 s, respectively, which is impressive. Tese num- bers are interesting, because the developers of OPTICS (Ankerst et al. [38]) reported that OPTICS was consist- ently 1.6 times slower than DBSCAN. However, their observations may not be applicable to today’s computers since they assessed it in their computer 20 years ago. It should be noted that the time complexity of both algo- rithms is dependent on ε and the underlying data struc- ture, and the general runtime complexity is the same for Fig. 2 Flowchart for assessing robustness using additive noise both: O(n*log(n)). Hence, to determine any consistent dif- ferences in execution time, the two algorithms must be repeatedly run on diferent datasets.

Fig. 3 Parameter choice in DBSCAN clustering: Epsilon plot for (a) SFC, and (b) DFC. The red dot refers to the fnal chosen epsilon value Rangaprakash et al. Brain Inf. (2020) 7:19 Page 8 of 13

Table 2 Success rate of clustering for each group and each Table 4 Separation index as a measure of OPTICS feature, for both DBSCAN and OPTICS robustness Success rate of clustering Row-wise Group-wise value of separation index Mean average % DBSCAN OPTICS Control EMCI LMCI AD

SFC % DFC % SFC % DFC % SFC 3.2914 3.6381 3.1946 3.2487 3.3432

Control 80 97.14 97.14 100 93.57 DFC 4.4227 3.8152 3.4321 4.1152 3.9463 EMCI 73.53 79.41 100 91.18 86.03 Higher value indicates better performance LMCI 64.71 82.35 82.35 91.18 80.15 AD 82.76 93.10 93.10 100 92.24 3.2 Separation index to evaluate robustness Mean 75 87.88 93.18 95.46 of reachability plot in OPTICS clustering Row-wise averages (last column) and column-wise averages (last row), shown in italics, provide summary statistics Te reachability plot determines the quality of OPTICS clustering, and hence we devised a measure called the separation index to evaluate the robustness of the reach- Table 3 SNR values obtained as a measure of robustness ability plot. A higher value indicates superior robustness. SFC DFC Upon evaluating the measure with both SFC and DFC DBSCAN 55 46 features (see Table 4), we found that DFC features pro- OPTICS 34 21 vided a higher value of separation index, which in turn indicates that DFC, compared to SFC features, is a more Lower value indicates better performance robust measure for performing clustering using OPTICS. 3.1 Assessing robustness using additive noise Tis result corroborates with the result obtained from As noted earlier, higher cluster purity does not necessar- R-CLAN regarding robustness using additive noise, ily imply its robustness to noise or outliers. We devised wherein the DFC features resulted in higher robustness a novel technique to assess the robustness of clustering compared to the SFC features. Tese fndings further (R-CLAN). A lower SNR value identifed by R-CLAN reiterate that OPTICS is a better clustering algorithm indicates that the clustering method is more robust, than DBSCAN for unsupervised clustering of fMRI con- because it would have delivered the same clustering nectivity features obtained from subjects with cognitive performance even at higher noise levels. We applied impairment, and that DFC features would result in better this technique to clustering of both SFC and DFC fea- clustering performance than SFC features. tures using both DBSCAN and OPTICS (see Table 3). 4 Discussion Clearly, OPTICS clustering was more robust to noise than DBSCAN since lower SNR values are indicative of In this work, density-based clustering was performed higher robustness. We obtained higher robustness with on static and dynamic functional connectivity features both SFC and DFC features using OPTICS, indicating obtained from fMRI data of subjects with cognitive that OPTICS could deliver the same level of clustering impairment. Te data consisted of subjects with early and performance even at higher noise levels. OPTICS could late mild cognitive impairment, Alzheimer’s disease and withstand noise levels of 125–315 times (or 21–25 dB) matched aged healthy controls. DBSCAN and OPTICS more than what DBSCAN could. It was also observed clustering techniques were applied to SFC as well as that DFC features were more robust to noise compared DFC features obtained from fMRI data. Since the input to SFC features, and the best performance was delivered data came from four diagnostic categories, we expected by OPTICS using DFC features, with an SNR of 21 dB. to identify four clusters using DBSCAN and OPTICS, DFC features could withstand about 20 times (or 13 dB) although the algorithms were not biased with this infor- more noise than SFC features using OPTICS. With these mation a priori. Upon clustering, we found that both observations, we concluded that OPTICS is a more DBSCAN and OPTICS resulted in four clusters, obtained robust clustering technique than DBSCAN, and that DFC in a blind unsupervised manner. Tis shows that these features could result in more robust clustering perfor- techniques were able to detect inherent disease clusters mance than SFC features. Tese fndings also corrobo- from neuroimaging data without any supervision. rate with clustering results presented earlier, wherein We found that higher cluster purities were obtained OPTICS and DFC resulted in better cluster purities than with OPTICS clustering compared to DBSCAN, and DBSCAN and SFC. Our fndings attribute both superior that DFC features always resulted in superior cluster- performance and higher robustness to OPTICS cluster- ing performance compared to SFC features. Further- ing using DFC features. more, the same trend was observed with the measures Rangaprakash et al. Brain Inf. (2020) 7:19 Page 9 of 13

of robustness and separation index. OPTICS clustering is a measure of how much the quality of data, evaluated combined with DFC features resulted in highest perfor- by SNR, must change before the clustering changes the mance as well as most robustness to noise. Previous work subject classes. Te ‘diferent’ data here is not a com- by Jia et al. [46] reported a similar superior performance pletely new set of data, but the same data changed by of DFC over SFC, wherein they determined that DFC noise. Besides the input parameters, feature selection features explained signifcantly more variance in human and other algorithmic characteristics, an unsupervised behavior than SFC. Another report by Jin et al. [47] found learning algorithm would be sensitive to outliers in the that DFC features have better ability in predicting psychi- data as well as underlying data quality. Since OPTICS atric disorders (such as post-traumatic stress disorder) and DBSCAN are already designed to detect outliers [37, compared to SFC features. Our fndings corroborate with 38], our robustness measures are a novel way of evalu- these previous reports. ating their sensitivity to data quality. According to data Referring to Table 2, the average clustering perfor- mining literature, this is the diference between robust- mance was found to be higher in control (93.57%) and ness to class noise (identifying outliers [56]), and to AD groups (92.24%), as compared to the intermediate attribute noise (errors in the feature values themselves EMCI (86.03%) and LMCI (80.15%) groups. Tis indi- [57]). Attribute noise tends to be more difcult to detect cates a rather expected trend wherein the control (com- and eliminate than class noise [58]; hence, it is impor- pletely healthy) and AD (completely ill) groups are on the tant that clustering is able to work well in the presence extreme, resulting in less outliers, as compared to EMCI of high attribute noise, as it is common in fMRI data and LMCI groups that form the mid-band of the spec- even after pre-processing and noise reduction [39]. To trum, resulting in more outliers and lower performance. the best of our knowledge, little research has been done A novel technique using additive noise (called in determining the level of attribute noise in fMRI data R-CLAN) was devised by us to assess the robustness that either clustering or a supervised classifer can with- of DBSCAN and OPTICS clustering. Table 3 shows stand before losing accuracy. Using our robustness meas- that OPTICS clustering was more robust to noise than ures, results indicate that OPTICS and DFC features are DBSCAN for both SFC and DFC feature sets, demon- the best combination of clustering method and input strating that OPTICS would be able to withstand higher features, respectively, to use in the presence of attribute noise levels than DBSCAN. In addition, clustering with noise, as compared to DBSCAN and SFC features. DFC features resulted in superior clustering perfor- During our evaluation of robustness, we modeled the mance as well as superior robustness to noise than clus- noise present in SFC and DFC features using additive tering with SFC features. Static connectivity is popular white Gaussian noise. It is notable that modeling noise in neuroimaging research and is extensively used, while in fMRI with a white Gaussian distribution is well doc- dynamic connectivity is a newer technique that has umented [40, 41], supported by the fact that non-bio- gained traction more recently [42]. Both SFC and DFC logical noise comes from various independent sources provide characteristically distinct information. While [40], which tend to resemble white noise. In addition, the former provides the strength of connectivity over the Chen and Tyler [40] showed that the power spectrum entire fMRI scan, the latter provides the temporal vari- of non-biological noise approached the fat line charac- ability of connectivity (i.e. change in connectivity over teristic of a white noise spectrum, and hence could be time). Recent studies have demonstrated the unique and well-approximated by white Gaussian noise; however, superior properties of DFC over SFC [46, 55]. A compar- biological noise was non-Gaussian. Other works indi- ison of SFC and DFC in their ability to identify hidden cate that the Gaussian noise assumption is appropriate structures in the data is one of the novel contributions of for data with high SNR values, but noise approaches our work, wherein DFC was found to be better and more a Rician distribution at lower SNR values [39, 59, 60]. robust than SFC. Tis is important, because the research Tus, there may be a weakness in our current model community still largely prefers to not look beyond static of noise since it does not consider non-white sources connectivity. We hope our fndings encourage research- of noise that can corrupt features. When developing a ers to incorporate dynamic connectivity analysis among noise reduction method on fMRI data obtained from their research strategies. Alzheimer’s disease patients, Garg et al. [41] used addi- Traditionally, supervised classifers are termed ‘robust’ tive white Gaussian noise and Rician noise separately when the testing error is close to the training error–that to model diferent levels of noise in artifcial and real- is, the classifer is able to perform well even on diferent world data [41]. Likewise, future work could compare datasets on which it is previously not trained. Our def- additive Rician noise and white Gaussian noise within nition of robustness works in a similar manner: by add- the R-CLAN framework. Te response to clustering ing noise to the underlying data, our form of robustness Rangaprakash et al. Brain Inf. (2020) 7:19 Page 10 of 13

with various levels of diferent types of attribute noise We used static and dynamic functional connectivity in could be investigated as well. this work. Te brain networks obtained from them can Te idea of measuring noise robustness of clustering also be used in a graph-theoretic framework and future techniques is not alien to the literature. For example, studies could attempt clustering of network properties to in the original work that presented OPTICS, a steep- obtain newer insights. ness value was used to determine at which point clus- Supervised algorithms such as support vector machine ters begin and end [38]. While it worked well in theory, (SVM) are popular in brain imaging. Backed by results its success was found to depend highly on the steep- of this study, we encourage researchers to consider den- ness value, an input parameter with no standard way of sity-based clustering methods in subject grouping and determining its appropriate value. In another method classifcation. Future work in this area could involve per- [61], similar to the peaks we used in fnding the separa- forming these analyses on various neurological and psy- tion index, all signifcant local maxima in the plot were chiatric disease conditions such as epilepsy, attention found, and signifcance was used to distinguish deep defcit hyperactivity disorder (ADHD), schizophrenia, valleys representing well-formed clusters from shallow autism spectrum disorder (ASD), post-traumatic stress regions that were simply noise. Te separation index is disorder (PTSD), etc. to determine if disease classifca- similar to these aforementioned methods, in that it is a tion fndings are consistently good across various clinical measure of how well defned each cluster is with respect populations. to the noise points surrounding it. However, it is a much Our main contributions are as follows: (i) we demon- simpler method and requires no human inputs. It could strated that unsupervised learning, which is not prone also be used in conjunction with the previous methods to overftting, could be used to determine inherent dis- and other cluster extracting methods to determine how ease clusters that provide performances comparable to well-separated a specifc cluster is from the points that supervised learning. (ii) We demonstrated that density- form its boundaries on the reachability plot. In addition, based clustering methods, which require minimal input separation index could be used as a measure of relative parameters, perform satisfactorily, and that the OPTICS density of each cluster, since higher separation indices technique is the suitable choice for fMRI connectivity usually correspond to deeper valleys, which in turn indi- data. (iii) We proposed two novel robustness assessment cates high-density regions within the dataset. techniques and demonstrated that OPTICS clustering It is interesting to note that DBSCAN and OPTICS is a superior and noise-robust technique for fMRI con- were found to be just as competitive in classifying Alz- nectivity data. (iv) For the frst time in the literature, we heimer’s disease patients as traditional supervised learn- assessed and compared the ability of static and dynamic ing classifers [62]. Tese two density-based clustering connectivity features in identifying inherent disease algorithms automatically detect outliers as part of their clusters and found dynamic connectivity features to be function, leaving us able to test for robustness to data superior and more robust. Tis is a signifcant fnding quality–something that is difcult to do with supervised given that the research community is still largely inclined classifers since outlier noise is more harmful to their towards the continued use of static connectivity alone. performance [58]. In addition, unsupervised algorithms such as DBSCAN and OPTICS can be used in applica- 5 Conclusion tions beyond just classifcation. Tey can also be used Unsupervised learning algorithms present some key to determine hidden disease states or sub-states that are theoretical advantages compared to supervised learn- often undetected in traditional diagnostic classifcation, ing algorithms (e.g. a priori diagnostic labeling is not future prediction of diagnostic status of new subjects required, and they possess the potential to detect based on current clusters [63] and hypothesis genera- unknown data structures). Our work presented and con- tion and testing, to name a few. For example, with cer- trasted two relatively new methods to perform unsu- ebrospinal fuid data, structural MRI and FDG-PET scans pervised clustering on fMRI data in a relatively large as features, an earlier study used clinical dataset. We briefy described and assessed the on healthy controls to identify subgroups within these performance of two density-based clustering algorithms: subjects that could later be susceptible to Alzheimer’s DBSCAN and OPTICS. Tese algorithms were used disease [64]. However, the number of clusters had to be to cluster three stages of cognitive impairment (EMCI, chosen through visual assessment prior to clustering. LMCI and AD) and matched healthy controls using static Our results indicate that similar experiments could be and dynamic functional connectivity features. DBSCAN performed using density-based clustering methods that was found to be relatively more sensitive to noise and less require few input parameters, and no requirement to precise, whereas OPTICS accurately identifed all four provide the number of clusters. groups and was more robust to noise as measured from Rangaprakash et al. Brain Inf. (2020) 7:19 Page 11 of 13

our proposed R-CLAN and separation index robustness Funding The authors acknowledge support from the Auburn University MRI Research measures. OPTICS clustering using DFC features was Center. The funders had no role in study design, data collection and analysis, found to be more dependable than DBSCAN and SFC decision to publish, or preparation of the manuscript. features. With the superiority of DFC and OPTICS, we Availability of data and materials encourage researchers to incorporate dynamic connec- The datasets used in this work were taken from the publicly available ADNI tivity analysis among their research strategies and hope database, which is available for download from their website (https​://www. that we have motivated the community sufciently to loni.ucla.edu/ADNI). consider employing OPTICS clustering for subject clas- Competing interests sifcation and identifying hidden disease clusters. The authors declare that they have no competing interests.

Supplementary information Author details 1 Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Supplementary information accompanies this paper at https​://doi. Hospital, Charlestown, MA, USA. 2 Department of Radiology, Harvard Medi- org/10.1186/s4070​8-020-00120​-2. cal School, Boston, MA, USA. 3 Division of Health Sciences and Technology, Harvard University and Massachusetts Institute of Technology, Cambridge, MA, USA. 4 Division of Engineering Science, Faculty of Applied Science & Engineer- ing, University of Toronto, Toronto, ON, Canada. 5 Department of Electrical Additional fle 1. Supplemental information. Communication Engineering, Indian Institute of Science, Bangalore, India. 6 AU MRI Research Center, Department of Electrical and Computer Engineer- ing, Auburn University, 560 Devall Dr, Suite 266D, Auburn, AL 36849, USA. Abbreviations 7 Department of Psychological Sciences, Auburn University, Auburn, AL, USA. AD: Alzheimer’s disease; ADNI: Alzheimer’s disease neuroimaging initiative; 8 Alabama Advanced Imaging Consortium, University of Alabama Birming- DBSCAN: Density-based spatial clustering of applications with noise; DFC: ham, Alabama, USA. 9 Center for Health Ecology and Equity Research, Auburn Dynamic functional connectivity; EMCI: Early mild cognitive impairment; FC: University, Auburn, AL, USA. 10 Center for Neuroscience, Auburn University, Functional connectivity; fMRI: Functional magnetic resonance imaging; LMCI: Auburn, AL, USA. 11 School of Psychology, Capital Normal University, Beijing, Late mild cognitive impairment; MCI: Mild cognitive impairment; OPTICS: China. 12 Key Laboratory for Learning and Cognition, Capital Normal University, Ordering points to identify the clustering structure; R-CLAN: Recursive-cluster- Beijing, China. 13 Department of Psychiatry, National Institute of Mental Health ing using additive-noise; rs-fMRI: Resting-state functional magnetic resonance and Neurosciences, Bangalore, India. imaging; SFC: Static functional connectivity; SNR: Signal-to-noise ratio; SVM: Support vector machine. Received: 16 October 2020 Accepted: 29 October 2020 Acknowledgements Data used in this article were obtained from the Alzheimer’s Disease Neuroim- aging Initiative (ADNI) database (adni.loni.usc.edu). Investigators within ADNI contributed to design and implementation of ADNI and provided data but did not participate in analysis or writing of this report. Complete listing of ADNI References investigators: https​://adni.loni.usc.edu/wp-conte​nt/uploa​ds/how_to_apply​/ 1. Wang Y, Chattaraman V, Kim HJ, Deshpande G (2015) Predicting purchase ADNI_Ackno​wledg​ement​_List.pdf. Data collection and sharing for this work decisions based on spatio-temporal functional MRI features using was funded by ADNI (National Institutes of Health Grant U01 AG024904) and machine learning. IEEE Transact Autonom Mental Devel 7(3):248–255 DOD ADNI (Department of Defense award number W81XWH-12-2-0012). 2. Mitchell TM, Hutchinson R, Niculescu RS, Pereira F, Wang X, Just M, New- ADNI is funded by the National Institute on Aging, the National Institute of man S (2004) Learning to Decode Cognitive States from Brain Images. Biomedical Imaging and Bioengineering, and through generous contributions Mach Learn 57(1–2):145–175 from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discov- 3. Clark IA, Niehaus KE, Duf EP, Di Simplicio MC, Cliford GD, Smith SM, ery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Mackay CE, Woolrich MW, Holmes EA (2014) First steps in using machine Company; CereSpir, Inc.; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and learning on fMRI data to predict intrusive memories of traumatic flm Company; EuroImmun; F. Hofmann-La Roche Ltd and its afliated company footage. Behav Res Ther 62:37–46 Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immu- 4. Norman KA, Polyn SM, Detre GJ, Haxby JV (2006) Beyond mind-reading: notherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical multi-voxel pattern analysis of fMRI data. Trends Cognit Sci 10(9):424–430 Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso 5. Demirci O, Clark VP, Magnotta VA, Andreasen NC, Lauriello J, Kiehl KA, Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pearlson GD, Calhoun VD (2008) A review of challenges in the use of Pharmaceuticals Corporation; Pfzer Inc.; Piramal Imaging; Servier; Takeda Phar- fMRI for disease classifcation / characterization and a projection pursuit maceutical Company; and Transition Therapeutics. The Canadian Institutes of application from Multi-site fMRI schizophrenia study. Brain Imag Behav Health Research is providing funds to support ADNI clinical sites in Canada. 2(3):147–226 Private sector contributions are facilitated by the Foundation for NIH (www. 6. Davatzikos C, Ruparel K, Fan Y, Shen DG, Acharyya M, Loughead JW, Gur fnih.org). The grantee is the Northern California Institute for Research and RC, Langleben DD (2005) Classifying spatial patterns of brain activity with Education, and the study is coordinated by Alzheimer’s Disease Cooperative machine learning methods: application to lie detection. Neuroimage Study at the University of California, San Diego. ADNI data are disseminated by 28(3):663–668 the Laboratory for Neuro Imaging at the University of Southern California. The 7. Deshpande G, Libero L, Sreenivasan KR, Deshpande H, Kana RK (2013) authors thank Katherine Buchanan, Auburn University for helping in language Identifcation of neural connectivity signatures of autism using machine editing. learning. Front Hum Neurosci 7:670 8. Deshpande G, Wang P, Rangaprakash D, Wilamowski B (2015) Fully con- Authors’ contributions nected cascade artifcial neural network architecture for attention defcit DR, NDD and GD were involved in the conception of this work. DR, TO and hyperactivity disorder classifcation from functional magnetic resonance GD analyzed and interpreted the data. All authors contributed to manuscript imaging data. IEEE Transact Cybernet 45(12):2668–2679 writing. All authors read and approved the fnal manuscript. 9. Libero LE, DeRamus TP, Lahti AC, Deshpande G, Kana RK (2015) Multi- modal neuroimaging based classifcation of Autism Spectrum Disorder using anatomical, neurochemical and white matter correlates. Cortex 66:46–59 Rangaprakash et al. Brain Inf. (2020) 7:19 Page 12 of 13

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